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ON QUADRATIC EXTENSIONS OF NUMBER FIELDS AND IWASAWA INVARIANTS FOR BASIC _3-EXTENSIONS

机译:关于基本_3-扩展数的场的二次扩展和IWAASAWA不变量

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摘要

Let Z_3 be the ring of 3-adic integers. For each number field F,let F_∞,3 denote the basic Z_3-extension over F; let λ_3(F) denote respectively the Iwasawa λ- and μ-invariants of F_∞,3/F. Here a number field means a finite extension over the rational field Q contained in the complex field C;F ∽C,[F:Q] < ∞,Now let k be a number field. Let L_2+ denote the infinite set of totally imaginary quadratic extensions in C over k (so that L_ coincides with the set L~- in the text when k is totally real );let L_+ denote the infinite set of quadratic extensions in C over k in which every infinite place of k splits (so that L_+ coincides with the set L~+ in the text when k is totally real ). After studying the distribution of certain quadrtic extensions over k, that of certain cubic that ,if k is totally real, then a subset of {K ∈L_|λ_3(K)=λ_(K),μ_3(K)=μ(K)=μ _3(K)} has an explicit positive density in L_. The paper also proves that a subset of {L ∈ L_+|λ_3(L)=μ_3(L)=0} has an explicit positive density in L_+ if 3 does not divide the class number of k but is divided by only one prime ideal of k. Some consequences of the above results are added in the last part of the paper.
机译:令Z_3为3-adic整数的环。对于每个数字字段F,令F_∞,3表示F上的基本Z_3扩展;令λ_3(F)分别表示F_∞,3 / F的Iwasawaλ和μ不变量。此处,数字字段表示对包含在复数字段C; F∽C,[F:Q] <∞中的有理字段Q的有限扩展,现在让k为数字字段。令L_2 +表示C上k上的全虚二次扩展的无穷大集合(因此当k完全为实时,L_与本文中的集合L〜-一致);而L_ +表示C上k的二次虚扩展的无穷大集合。其中k的每个无穷大位置都会分裂(因此当k完全为实时,L_ +与文本中的集合L〜+重合)。在研究了k上的某些二次扩展的分布之后,如果k完全为实,则确定某个三次方的分布,则{K∈L_|λ_3(K)=λ_(K),μ_3(K)=μ(K )=μ_3(K)}在L_中具有明确的正密度。本文还证明,如果3不将k的类别数除而仅除以1,则{L∈L_ + |λ_3(L)=μ_3(L)= 0}的子集在L_ +中具有明确的正密度。 k的最理想。以上结果的一些后果将添加到本文的最后一部分。

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